东南大学：《固体力学基础》课程教学课件（英文讲稿）03 Stress Measures

Stress Measuresmi@se.ed.cn

Stress Measures

Outline·Bodyand surfaceforces（体力与面力）Traction/stressvector（应力矢量）Cauchystresstensor（柯西应力张量）Tractiononobliqueplanes（斜面上的应力）Differentstressmeasures（应力度量）Stressmeasuresfor infinitesimaldeformation（小变形应力度量）Principal stressesanddirections（主应力与主方向），Octahedral stresses（八面体应力）· Hydrostatic, deviatoric and von Mises effective stresses (平均应力、偏应力、米泽斯等效应力）Conservationoflinearmomentum（线动量守恒）Conservationofangularmomentum（角动量守恒）·Rateof workdoneby stresses（应力做功的速率）2

Outline • Body and surface forces（体力与面力） • Traction/stress vector（应力矢量） • Cauchy stress tensor （柯西应力张量） • Traction on oblique planes（斜面上的应力） • Different stress measures（应力度量） • Stress measures for infinitesimal deformation（小变形应力 度量） • Principal stresses and directions（主应力与主方向） • Octahedral stresses（八面体应力） • Hydrostatic, deviatoric and von Mises effective stresses（平 均应力、偏应力、米泽斯等效应力） • Conservation of linear momentum（线动量守恒） • Conservation of angular momentum（角动量守恒） • Rate of work done by stresses（应力做功的速率） 2

Body and SurfaceForces External loads include body and surface forcesP3Body Forces: F(x)PSurfaceforcesForces are vectors (unit: N)F= Fe, +Fe2 +Fe, = Fe Often interpreted in terms of density: body force densityand surface force densityT"(x)dSFRF(x)dVS3

Body and Surface Forces • External loads include body and surface forces. Surface forces • Forces are vectors (unit: N) F e e e e     F F F F 1 1 2 2 3 3 i i • Often interpreted in terms of density: body force density and surface force density 3

Traction/Stress VectornAFnLARDeformedOriginalconfigurationconfigurationGiven AF as the force transmitted across △A, a stresstraction vector can be defined asAFT() (x,n)= limM->0 AUnits: Pa (N/m2), 1 MPa = 106 Pa, 1 GPa = 109 PaDecomposition of the traction vectorT(") (x,n) =o n+tt=o n+t't'+t"t"4

Tn σ  Units: Pa (N/m2 ), 1 MPa = 106 Pa, 1 GPa = 109 Pa. • Given ΔF as the force transmitted across ΔA, a stress traction vector can be defined as F A n 4 Traction/Stress Vector     0 , lim  A A    n F T x n    ,                n T x n n t n t t • Decomposition of the traction vector

Cauchy Stress Tensorr(n) (T(x,n=e)=oe+twe,+txe.x.n=+te.+=Te+o.e.OOxzxyVZZXV2A福0zy7Txy0xvonKarmanNotation5

5 Cauchy Stress Tensor    ,      x x x xy y xz z    n T x n e e e e    ,     y yx x y y yz z     n T x n e e e e    , z zx x zy y z z        n T x n e e e e x xy xz ij yx y yz zx zy z                          von Karman Notation